14 1 Deformation and Fracture of Perfect Crystals 1.1.1.2 Ab initio Methods Nowadays,so-called ab initio (or first principle)approaches enable us to compute the crystal energy in a very accurate manner.This particularly holds for single crystals of pure elements or compounds,crystals with periodical arrangements of atoms of various kinds and also for local crystal defects. These methods utilize the density functional theory 37,38 in which the problem of many interacting electrons is transformed into a study of single electron motion in an effective potential.A brief description of principles of such methods is presented in Appendix A. Most probably,the first ab initio calculation of the uniaxial IS oiut was that of Esposito et al.39 for the copper crystal.However,these authors did not perform relaxations of atomic positions inside the loaded crystal in directions perpendicular to the loading axis(Poisson's type of expansion or contraction).Probably the first ab initio simulation of a tensile test that included the relaxation in perpendicular directions to the loading axis was performed by Price et al.[40]for TiC along the 001]axis.Later,oiut was calculated for [001]and [111]loading axes for a variety of cubic crystals by Sob et al.[41-43].Kitagawa and Ogata [44,45]studied the tensile IS of Al and AlN but also did not include the Poisson's contraction.Further calculations of oiut, performed for a-SiC,diamond,Si,Ge,Mo,Nb and SigN4,have already taken that effect into account by allowing a transversal relaxation of atoms [46-51. The values Tis,b of the shear IS were first calculated by Paxton et al.[32 for V,Cr,Nb,Mo,W,Al,Cu and Ir.The values Tis calculated according to the model of a uniform shear(see Figure 1.3)were later reported by Moriarty et al.52,53 for Mo and Ta.These calculations did not include any relaxations. Recently,the relaxed values of Tis were calculated for many crystals such as TiC,TiN,HfC,Mo,Nb,Si,Al,Cu and W by groups of Morris et al.,Kitamura et al.and Pokluda et al.[54-58].In these models,the interplanar distance was relaxed towards the minimum energy during deformation.More advanced models also enabled relaxations of the arrangement of atomic positions within the planes [48,59-see also Sections 1.1.2 and 1.1.3. Since 1997,ab initio calculations of oiht have been performed by Pokluda et al.[60-62]and other authors (e.g.,63)).Since spherical symmetry was maintained during deformation,the relaxation procedures were not necessar- ily applied in these models. In the majority of older studies on IS,the deformation process was assumed to proceed in a stable manner until the applied stress reached its maximum value.This means that the crystal failed in the same mode in which it was originally deformed from the very beginning.However,this assumption was disputed in many works [64-66.Under both tensile and compressive load- ings,the shear stresses in some slip systems can exceed their critical values (corresponding to the related Tis)well before the stress reaches its maximum. This was also observed in tensile tests on whiskers [30,33.Indeed,some of the whiskers evidently failed by shear across some favourable crystallographic
14 1 Deformation and Fracture of Perfect Crystals 1.1.1.2 Ab initio Methods Nowadays, so-called ab initio (or first principle) approaches enable us to compute the crystal energy in a very accurate manner. This particularly holds for single crystals of pure elements or compounds, crystals with periodical arrangements of atoms of various kinds and also for local crystal defects. These methods utilize the density functional theory [37, 38] in which the problem of many interacting electrons is transformed into a study of single electron motion in an effective potential. A brief description of principles of such methods is presented in Appendix A. Most probably, the first ab initio calculation of the uniaxial IS σiut was that of Esposito et al. [39] for the copper crystal. However, these authors did not perform relaxations of atomic positions inside the loaded crystal in directions perpendicular to the loading axis (Poisson’s type of expansion or contraction). Probably the first ab initio simulation of a tensile test that included the relaxation in perpendicular directions to the loading axis was performed by Price et al. [40] for TiC along the [001] axis. Later, σiut was calculated for [001] and [111] loading axes for a variety of cubic crystals by Sob ˇ et al. [41–43]. Kitagawa and Ogata [44,45] studied the tensile IS of Al and AlN but also did not include the Poisson’s contraction. Further calculations of σiut, performed for α-SiC, diamond, Si, Ge, Mo, Nb and Si3N4, have already taken that effect into account by allowing a transversal relaxation of atoms [46–51]. The values τis,b of the shear IS were first calculated by Paxton et al. [32] for V, Cr, Nb, Mo, W, Al, Cu and Ir. The values τ ∗ is calculated according to the model of a uniform shear (see Figure 1.3) were later reported by Moriarty et al. [52,53] for Mo and Ta. These calculations did not include any relaxations. Recently, the relaxed values of τis were calculated for many crystals such as TiC, TiN, HfC, Mo, Nb, Si, Al, Cu and W by groups of Morris et al., Kitamura et al. and Pokluda et al. [54–58]. In these models, the interplanar distance was relaxed towards the minimum energy during deformation. More advanced models also enabled relaxations of the arrangement of atomic positions within the planes [48, 59] – see also Sections 1.1.2 and 1.1.3. Since 1997, ab initio calculations of σiht have been performed by Pokluda et al. [60–62] and other authors (e.g., [63]). Since spherical symmetry was maintained during deformation, the relaxation procedures were not necessarily applied in these models. In the majority of older studies on IS, the deformation process was assumed to proceed in a stable manner until the applied stress reached its maximum value. This means that the crystal failed in the same mode in which it was originally deformed from the very beginning. However, this assumption was disputed in many works [64–66]. Under both tensile and compressive loadings, the shear stresses in some slip systems can exceed their critical values (corresponding to the related τis) well before the stress reaches its maximum. This was also observed in tensile tests on whiskers [30, 33]. Indeed, some of the whiskers evidently failed by shear across some favourable crystallographic
1.1 Ideal Strength of Solids 15 Figure 1.3 Scheme of the uniform shear applied to a perfect crystal plane resulting in an atomically smooth fracture surface.Thus,the definition of IS as a maximum attainable stress along the deformation path was assigned to a stress related to the first onset of instability.Many stability studies were based on an analysis of the elastic response of crystals subjected to small homogeneous deformations.Such analyses of the mechanical stability,based on calculations of independent elastic moduli,led to a significant decrease in computed values of IS.This was the case for 001 uniaxial loading in Al,Nb and Cu [49,67,68.More attention to this issue will be paid in Sections 1.1.2 and1.1.3. 1.1.1.3 Other Advanced Methods Besides ab initio approaches several other methods were also utilized for IS computations.Although these methods lie beyond the scope of this book,it is worthwhile making a brief comment on the topic. In the last 20 years,very sophisticated semi-empirical methods such as many-body potentials of Finnis-Sinclair type 69],embedded atom method 69]or bond-order potentials [70]were used for analyses of extended crystal defects [71-73].These concepts represent a hybrid between empirical and ab initio approaches.At present,the ab initio methods are only capable of giving a sufficiently precise prediction of the mechanical behaviour in very simple cases (but still far enough from the unstressed equilibrium state).This is the reason why,starting from the early 1990s,they are used for the calibration of semi-empirical potentials [42]. A further step towards identification of the initial onset of an instability was made by studies on the phonon spectra of crystal states along investi- gated deformation paths.This approach has further reduced the calculated values of IS [74].The phonons are quasi particles used to express a particle aspect of lattice vibrations.They play a major role,e.g.,in the theory of both
1.1 Ideal Strength of Solids 15 a s b Figure 1.3 Scheme of the uniform shear applied to a perfect crystal plane resulting in an atomically smooth fracture surface. Thus, the definition of IS as a maximum attainable stress along the deformation path was assigned to a stress related to the first onset of instability. Many stability studies were based on an analysis of the elastic response of crystals subjected to small homogeneous deformations. Such analyses of the mechanical stability, based on calculations of independent elastic moduli, led to a significant decrease in computed values of IS. This was the case for [001] uniaxial loading in Al, Nb and Cu [49, 67, 68]. More attention to this issue will be paid in Sections 1.1.2 and 1.1.3. 1.1.1.3 Other Advanced Methods Besides ab initio approaches several other methods were also utilized for IS computations. Although these methods lie beyond the scope of this book, it is worthwhile making a brief comment on the topic. In the last 20 years, very sophisticated semi-empirical methods such as many-body potentials of Finnis–Sinclair type [69], embedded atom method [69] or bond-order potentials [70] were used for analyses of extended crystal defects [71–73]. These concepts represent a hybrid between empirical and ab initio approaches. At present, the ab initio methods are only capable of giving a sufficiently precise prediction of the mechanical behaviour in very simple cases (but still far enough from the unstressed equilibrium state). This is the reason why, starting from the early 1990s, they are used for the calibration of semi-empirical potentials [42]. A further step towards identification of the initial onset of an instability was made by studies on the phonon spectra of crystal states along investigated deformation paths. This approach has further reduced the calculated values of IS [74]. The phonons are quasi particles used to express a particle aspect of lattice vibrations. They play a major role, e.g., in the theory of both
16 1 Deformation and Fracture of Perfect Crystals thermal and electric conductivity.However,they can also serve as an indica- tor of the lattice instability related to so-called soft phonon modes,at which the phonon frequency becomes an imaginary number.Such instabilities are responsible for various structural transitions [75,76 and,in general,they can be understood as an irreversible non-uniform (heterogeneous)distortion of a crystal.In this aspect,phonon analysis represents a generalization of elastic stability analysis since any observed elastic instability corresponds to a soft phonon mode with an infinite wavelength. Once the crystal becomes unstable,it will follow a trajectory in a con- figurational space that will eventually violate the harmonic approximation inherent in the phonon calculation.In order to find such trajectories,molec- ular dynamic (MD)methods can be utilized.These approaches account for variations of the unit cell shape as well as the positions of constituent atoms. MD methods are probably the most promising tools for an investigation of the system stability and eventual structure evolutions during spontaneous structural transitions.They can give a sufficient number of degrees of free- dom to studied systems and,furthermore,they can also incorporate finite temperatures and pressures thus bringing the simulations closer to reality. However,certain limitations related to computational capacity hinder them from wider applications.Present MD methods are mostly based on empirical or semi-empirical interatomic potentials.Some results obtained by means of these methods are mentioned in Section 1.2. Let us finally note that the most sophisticated methods,including so-called correlated electron-ion dynamics,are even more computationally demanding. On the other hand,they may represent a reliable tool for atomistic simula- tions in the near future 77. 1.1.2 Calculation Principles When applying atomistic approaches to a particular crystalline system,the dependence of its total energy on the deformation state constitutes the main output. In order to describe the deformation of a crystalline system,it is necessary to define appropriate deformation parameters (strain variables).However, there is no unique way of defining a set of parameters which would provide a measure of a pure finite strain related to the crystal reference state.Therefore, one can find several different definitions used in the technical literature.A homogeneous strain of a crystal can be specified,e.g.,by any six parameters that define a primitive cell.Some authors use lengths of cell edges (ai)and their included angles (a:)66,78-80].Although the deformation is described by changes of these parameters rather than themselves,they are also widely used in stability analyses.Such a natural set of variables is sometimes called Milstein's variables [79]
16 1 Deformation and Fracture of Perfect Crystals thermal and electric conductivity. However, they can also serve as an indicator of the lattice instability related to so-called soft phonon modes, at which the phonon frequency becomes an imaginary number. Such instabilities are responsible for various structural transitions [75,76] and, in general, they can be understood as an irreversible non-uniform (heterogeneous) distortion of a crystal. In this aspect, phonon analysis represents a generalization of elastic stability analysis since any observed elastic instability corresponds to a soft phonon mode with an infinite wavelength. Once the crystal becomes unstable, it will follow a trajectory in a con- figurational space that will eventually violate the harmonic approximation inherent in the phonon calculation. In order to find such trajectories, molecular dynamic (MD) methods can be utilized. These approaches account for variations of the unit cell shape as well as the positions of constituent atoms. MD methods are probably the most promising tools for an investigation of the system stability and eventual structure evolutions during spontaneous structural transitions. They can give a sufficient number of degrees of freedom to studied systems and, furthermore, they can also incorporate finite temperatures and pressures thus bringing the simulations closer to reality. However, certain limitations related to computational capacity hinder them from wider applications. Present MD methods are mostly based on empirical or semi-empirical interatomic potentials. Some results obtained by means of these methods are mentioned in Section 1.2. Let us finally note that the most sophisticated methods, including so-called correlated electron-ion dynamics, are even more computationally demanding. On the other hand, they may represent a reliable tool for atomistic simulations in the near future [77]. 1.1.2 Calculation Principles When applying atomistic approaches to a particular crystalline system, the dependence of its total energy on the deformation state constitutes the main output. In order to describe the deformation of a crystalline system, it is necessary to define appropriate deformation parameters (strain variables). However, there is no unique way of defining a set of parameters which would provide a measure of a pure finite strain related to the crystal reference state. Therefore, one can find several different definitions used in the technical literature. A homogeneous strain of a crystal can be specified, e.g., by any six parameters that define a primitive cell. Some authors use lengths of cell edges (ai) and their included angles (αi) [66, 78–80]. Although the deformation is described by changes of these parameters rather than themselves, they are also widely used in stability analyses. Such a natural set of variables is sometimes called Milstein’s variables [79]
1.1 Ideal Strength of Solids 17 A simple description of any deformation that is used throughout this work employs a general rule on how to change position of an arbitrary point (e.g., atomic position)within the chosen coordinate system.This rule is expressed by a transformation matrix (deformation gradient)which is also called the Jacobian matrix J 81.The practical advantage of such a description is ap- parent when one simulates a homogeneous deformation of a crystal through the deformation of its primitive cell.The primitive cell comprises both the motif and the set of vectors that determine the translational symmetry of a crystal.The transformation of vectors is then performed by multiplying by the Jacobian matrix. Let us consider a crystalline system in a reference state with the corre- sponding set of primitive translational vectors ar,br,cr.Applying a defor- mation described by J to the system,the set is transformed to a new set of vectors corresponding to the deformed state a=Jar;b=Jbr;c=JCr The tensor of a finite deformation (also called Lagrangian strain tensor) is defined according to the relation =0P小-0， where I means the identity matrix. An equivalent definition of the finite strain tensor uses a rule that describes changes of lattice points positions via the displacement vector u a-ar and the matrix eij oui/Oaj [82].When using the Einstein summation rule, the components of the finite strain tensor can then be written as 1(u+u+ Ouk Ouk n=2∂a (1.4) 8ai Dai dai The components nii refer to stretches and ni stands for shear strains if ij. A small deformation can also be depicted by a small strain tensor(known also as Euler strain,Green tensor or Cauchy infinitesimal strain): (+ (1.5) Hence,the finite strain differs from the small strain by the cross-term uu.Consequences of this difference with respect to the IS analysis are Bai Baj discussed in [83].As the tensor represents a symmetric part of the matrix eij, any infinitesimal transformation can be expressed by a linear combination of the tensor sij (describing pure deformation)and the antisymmetric part of the matrix eij 1 dui 2 da that represents the rotation as can be seen in Figure 1.4
1.1 Ideal Strength of Solids 17 A simple description of any deformation that is used throughout this work employs a general rule on how to change position of an arbitrary point (e.g., atomic position) within the chosen coordinate system. This rule is expressed by a transformation matrix (deformation gradient) which is also called the Jacobian matrix J [81]. The practical advantage of such a description is apparent when one simulates a homogeneous deformation of a crystal through the deformation of its primitive cell. The primitive cell comprises both the motif and the set of vectors that determine the translational symmetry of a crystal. The transformation of vectors is then performed by multiplying by the Jacobian matrix. Let us consider a crystalline system in a reference state with the corresponding set of primitive translational vectors ar, br, cr. Applying a deformation described by J to the system, the set is transformed to a new set of vectors corresponding to the deformed state a = Jar; b = Jbr; c = Jcr. The tensor of a finite deformation (also called Lagrangian strain tensor) is defined according to the relation ηˆ = 1 2 (JT J − I), where I means the identity matrix. An equivalent definition of the finite strain tensor uses a rule that describes changes of lattice points positions via the displacement vector u = a − ar and the matrix eij = ∂ui/∂aj [82]. When using the Einstein summation rule, the components of the finite strain tensor can then be written as ηij = 1 2 ∂ui ∂aj + ∂uj ∂ai + ∂uk ∂ai ∂uk ∂aj . (1.4) The components ηii refer to stretches and ηij stands for shear strains if i = j. A small deformation can also be depicted by a small strain tensor (known also as Euler strain, Green tensor or Cauchy infinitesimal strain): εij = 1 2 ∂ui ∂aj + ∂uj ∂ai . (1.5) Hence, the finite strain differs from the small strain by the cross-term ∂uk ∂ai ∂uk ∂aj . Consequences of this difference with respect to the IS analysis are discussed in [83]. As the tensor represents a symmetric part of the matrix eij , any infinitesimal transformation can be expressed by a linear combination of the tensor εij (describing pure deformation) and the antisymmetric part of the matrix eij ωij = 1 2 ∂ui ∂aj − ∂uj ∂ai that represents the rotation as can be seen in Figure 1.4
18 1 Deformation and Fracture of Perfect Crystals 十 Figure 1.4 Two-dimensional illustration of lattice distortion (simple shear)as a pure shear plus a rotation = As follows from Equations 1.4 and 1.5,both the strain tensors are sym- metric(even if J is generally asymmetric).Therefore,it is useful to use Voigt notation that reduces the number of indices of symmetric tensors according to the prescription 11→1 23→4 22→2 13→5 33→3 12→6. Thus,the second-rank tensor is reduced to a six-dimensional vector.Re- grettably,there are two different standards used in the literature.The first one exactly follows the above-mentioned substitution nij=na (where indices i,j run from 1 to 3 and the index a runs from 1 to 6)84,85]whereas the other one uses substitution nij =gn(1+ij)that leads to the following difference for the shear components:n4 2n23,n5 2m3 and n6 2n2 [82,86-89. The latter case is usually called standard Voigt notation.This simplifies the expression for energy expansion at Equation 1.7. When a crystal system is subjected to a deformation,its energy changes (in the case of a stable state it increases,of course).The crystal energy can be expanded into Taylor series with respect to the finite strain nij as U=Uo(V)+Vo+VCms+(n ) (1.6) where V represents a system volume,Cijki are elastic moduli and oij are the related stress tensor components.In classical continuum mechanics,Ciikt are usually called elastic constants [82.However,the analysis of crystal stability (see Section 1.1.2.2)requires calculation of Ciikl also at states far from equi- librium,where their values depend on the applied strain (or stress).In this book,therefore,the term elastic moduli is used for Cijki. The symmetry of Cijkt and oij with respect to interchange of indices (ij)and (enables us to use the Voigt notation for both of them
18 1 Deformation and Fracture of Perfect Crystals y x y x = + 2 2 2 Figure 1.4 Two-dimensional illustration of lattice distortion (simple shear) as a pure shear plus a rotation ˆe = ˆε + ˆω As follows from Equations 1.4 and 1.5, both the strain tensors are symmetric (even if J is generally asymmetric). Therefore, it is useful to use Voigt notation that reduces the number of indices of symmetric tensors according to the prescription 11 → 1 23 → 4 22 → 2 13 → 5 33 → 3 12 → 6. Thus, the second-rank tensor is reduced to a six-dimensional vector. Regrettably, there are two different standards used in the literature. The first one exactly follows the above-mentioned substitution ηij = ηα (where indices i, j run from 1 to 3 and the index α runs from 1 to 6) [84,85] whereas the other one uses substitution ηij = 1 2 ηα(1 + δij ) that leads to the following difference for the shear components: η4 = 2η23, η5 = 2η13 and η6 = 2η12 [82, 86–89]. The latter case is usually called standard Voigt notation. This simplifies the expression for energy expansion at Equation 1.7. When a crystal system is subjected to a deformation, its energy changes (in the case of a stable state it increases, of course). The crystal energy can be expanded into Taylor series with respect to the finite strain ηij as U = U0(V ) + V σijηij + 1 2 V Cijklηijηkl + O(η3), (1.6) where V represents a system volume, Cijkl are elastic moduli and σij are the related stress tensor components. In classical continuum mechanics, Cijkl are usually called elastic constants [82]. However, the analysis of crystal stability (see Section 1.1.2.2) requires calculation of Cijkl also at states far from equilibrium, where their values depend on the applied strain (or stress). In this book, therefore, the term elastic moduli is used for Cijkl. The symmetry of Cijkl and σij with respect to interchange of indices (i ↔ j) and (k ↔ l) enables us to use the Voigt notation for both of them